To level your alchemy, you need to make EXPENSIVE potions. The more expensive, the faster you will gain alchemy level.
This may help you.
Skyrim Alchemy Profitability Calculator
To find the ingredients, go around all the Alchemist shops in Skyrim and buy all the ingredients there. Then go down the list of each ingredients and make the most valuable potion from it. Always create MULTI EFFECTS potions as they are much more valuable.Go down you whole list then sell whatever potion you can to the Alchemist there. Move to another alchemist.
Also, get Fortify Alchemy enchanting gear, it will increase the effect and the value of the potion quite a bit.
If you go around all the Alchemists in Skyrim, it will take you more than 2 days of game time to do so. So when you are done with your last Alchemist, you can return to the first one, who would have restocked.
As you can see, you will get rich fast, increasing your Alchemy Skill as well as your Speech skill. At the end it comes to a point that you are making potions so expensive that Alchemist are not able to buy it anymore. My advice is to then get the Investor Perk in the speech tree. Then go invest in the General Good in Riverwood. This guy is specific because with Investor, he will get 10.000+ septims every 48h to buy stuff from you. Other merchant would only have an increase of 500 septims when you invest in them. A little bonus, because you took Investor, that also means that you have the Merchant perk, so all the merchants in game will be able to buy potions from you.
Don't forget to take the Thieve Stone, so you will gain level in Alchemy even faster.